To address the second question, mutually orthogonal vectors are linearly independent, but the converse is not always true.
To show this, consider a set of non-zero vectors $ \left\{ \mathbf{ x_1}, \mathbf{x_2}, ... , \mathbf{x_n} \right\} $ that are mutually orthogonal, that is, for any $a \neq b$, $\mathbf{x_a} \cdot \mathbf{x_b} = 0$.
Suppose the set of vectors is not linearly independent. By definition, that would imply that $\mathbf{x_1} = \sum_{i=2}^{n} c_i \mathbf{x_i}$. Now, since $\mathbf{x_1}$ is a non-zero vector, it must be true that $\mathbf{x_1} \cdot \mathbf{x_1} \neq 0$. However, $ \mathbf{x_1} \cdot \mathbf{x_1} = \mathbf{x_1} \cdot \sum_{i=2}^n c_i \mathbf{x_i} = \sum_{i=2}^n c_i \mathbf{x_1} \cdot \mathbf{x_i}$ as the dot product is distributive, and hence $\mathbf{x_1} \cdot \mathbf{x_1} = \sum_{i=2}^n c_i \cdot 0 = 0$ which is of course a contradiction, and so the earlier assumption that our vectors were not linearly independent must be incorrect.
Question: are the eigenvectors of a symmetric matrix mutually orthogonal? Are they linearly independent? Is this the same thing?
ReplyDeleteTo address the second question, mutually orthogonal vectors are linearly independent, but the converse is not always true.
ReplyDeleteTo show this, consider a set of non-zero vectors $ \left\{ \mathbf{ x_1}, \mathbf{x_2}, ... , \mathbf{x_n} \right\} $ that are mutually orthogonal, that is, for any $a \neq b$, $\mathbf{x_a} \cdot \mathbf{x_b} = 0$.
Suppose the set of vectors is not linearly independent. By definition, that would imply that $\mathbf{x_1} = \sum_{i=2}^{n} c_i \mathbf{x_i}$. Now, since $\mathbf{x_1}$ is a non-zero vector, it must be true that $\mathbf{x_1} \cdot \mathbf{x_1} \neq 0$. However,
$ \mathbf{x_1} \cdot \mathbf{x_1} = \mathbf{x_1} \cdot \sum_{i=2}^n c_i \mathbf{x_i} = \sum_{i=2}^n c_i \mathbf{x_1} \cdot \mathbf{x_i}$ as the dot product is distributive, and hence $\mathbf{x_1} \cdot \mathbf{x_1} = \sum_{i=2}^n c_i \cdot 0 = 0$ which is of course a contradiction, and so the earlier assumption that our vectors were not linearly independent must be incorrect.
Can we have the mark schemes for the solomon papers as well please? That would be really helpful, thanks!
ReplyDeleteI've added all the Solomon papers and mark schemes now. Enjoy
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ReplyDeleteHi Sir, I don't suppose you have any D2 notes that you could put up do you? thanks
ReplyDeleteI'm afraid not Kathryn. Have a look at the FMSP website?
ReplyDelete